Theorem sectrcl2 49382

Description: Reverse closure for section relations. (Contributed by Zhi Wang, 14-Nov-2025.)

Hypotheses

Ref	Expression
sectrcl.s	⊢ 𝑆 = (Sect‘𝐶)
sectrcl.f	⊢ (𝜑 → 𝐹(𝑋𝑆𝑌)𝐺)
sectrcl2.b	⊢ 𝐵 = (Base‘𝐶)

Assertion

Ref	Expression
sectrcl2	⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))

Proof of Theorem sectrcl2

Dummy variables 𝑥 𝑦 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sectrcl.f	. . . 4 ⊢ (𝜑 → 𝐹(𝑋𝑆𝑌)𝐺)
2		df-br 5101	. . . 4 ⊢ (𝐹(𝑋𝑆𝑌)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝑋𝑆𝑌))
3	1, 2	sylib 218	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝑋𝑆𝑌))
4		sectrcl2.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
5		eqid 2737	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
6		eqid 2737	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
7		eqid 2737	. . . . 5 ⊢ (Id‘𝐶) = (Id‘𝐶)
8		sectrcl.s	. . . . 5 ⊢ 𝑆 = (Sect‘𝐶)
9	8, 1	sectrcl 49381	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
10	4, 5, 6, 7, 8, 9	sectffval 17686	. . . 4 ⊢ (𝜑 → 𝑆 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑥)𝑓) = ((Id‘𝐶)‘𝑥))}))
11	10	oveqd 7385	. . 3 ⊢ (𝜑 → (𝑋𝑆𝑌) = (𝑋(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑥)𝑓) = ((Id‘𝐶)‘𝑥))})𝑌))
12	3, 11	eleqtrd 2839	. 2 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝑋(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑥)𝑓) = ((Id‘𝐶)‘𝑥))})𝑌))
13		eqid 2737	. . 3 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑥)𝑓) = ((Id‘𝐶)‘𝑥))}) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑥)𝑓) = ((Id‘𝐶)‘𝑥))})
14	13	elmpocl 7609	. 2 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝑋(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑥)𝑓) = ((Id‘𝐶)‘𝑥))})𝑌) → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))
15	12, 14	syl 17	1 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ⟨cop 4588   class class class wbr 5100  {copab 5162  ‘cfv 6500  (class class class)co 7368   ∈ cmpo 7370  Basecbs 17148  Hom chom 17200  compcco 17201  Idccid 17600  Sectcsect 17680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-sect 17683

This theorem is referenced by:  isinv2  49385  catcsect  49757